402 



SURFACE WAVES. 



[CHAP, ix 



surface, we have only to take the mean result of a series of lines 

 of pressure whose inclinations 6 are distributed uniformly between 

 the limits + JTT*. This result is expressed by a definite integral 

 whose interpretation would be difficult ; but a general idea of the 

 forms of the wave-ridges may be obtained by a process analogous 

 to that introduced by Huyghens in Physical Optics, viz. by tracing 

 the envelopes of the straight lines which represent them in the 

 component systems. It appears on reference to (21) that the 

 perpendicular distance p of any particular ridge from the origin 

 is given by 



p = (2-i)lT, 



where s is integral, and K = g/c* cos 2 6. The tangential polar equa 

 tion of the envelopes in question is therefore 



p = acos*0 (1), 



where, for consecutive crests or hollows, a differs by 27rc 2 /#. The 



forms of the curves are shewn in the annexed figure, traced from 

 the equations 



x =jocos 6 -^sin0= \ a (5 cos 6 cos SB), } 



[ (2). 



y = p sin + -A cos 6 = J a (sin 6 + sin 30) 



etc/ / 



* This artifice is taken from Lord Rayleigh s paper, cited on p. 393. 



